1. NETCOD 2013 June 7-9, Calgary Broadcast Erasure Channel with Feedback: the Two Multicast Case Algorithms and Bounds Efe Onaran 1, Marios Gatzianas 2 and Christina Fragouli 2 1 Department of Electrical and Electronics Engineering, Bilkent Univ., Turkey. 2 School of Computer and Communication Sciences, EPFL, Switzerland.

2. NETCOD 2013 June 7-9, Calgary Outline 2.A capacity outer bound and a conjecture. 1.Role of feedback on unicast and multicast traffic over BEC Motivation for multiple multicast sessions. 3.Exploiting feedback: performance/complexity tradeoff Optimal algorithm for 2 users per group. An algorithm that is asymptotically better than timesharing (as erasure prob 0) 4.Conclusions.

3. NETCOD 2013 June 7-9, Calgary Effect of feedback on BEC unicast capacity Fact: FB increases capacity region of BEC with multiple unicast sessions W/o FB (TS) With FB Benefit of FB-based scheme over TS (equal rates) for N users= Optimal FB scheme: 1.Transmit to D1. There exist packets missed by D1 and seen by D2. 2.Transmit to D2. There exist packets missed by D2 and seen by D1. 3.Transmit. 4.Also works with linear combs. increasing w.r.t. N Not only does FB help for multiple unicast traffic, it also yields increasing benefits vs number of users (especially for high ) SRC/TX D1 D2

4. NETCOD 2013 June 7-9, Calgary Effect of feedback on BEC multicast capacity Fact: For a single multicast session of users, FB offers no benefit. achievable through network coding Topic of this paper: what happens for multiple multicast sessions? Does FB offer any benefit? How does this benefit scale with number of users per group? SRC/TX Useful tools for 2 multicast problem: 2. An achievable scheme (benchmark): use TS between sessions 1. A multicast capacity outer bound use 2-unicast bound Inner bound (TS) Outer bound ?

5. NETCOD 2013 June 7-9, Calgary Our contribution For, outer bound is tight. For any, we propose a very simple FB-based algorithm that outperforms TS (especially for small ). All schemes we have tried so far approach TS performance-wise as. This suggests the following conjecture: Timesharing is asymptotically optimal as. We provide some evidence for this conjecture via a special case proof, specifically timesharing is asymptotically optimal if.

6. NETCOD 2013 June 7-9, Calgary Optimal scheme for N=2 (outline) The basic idea is taken from the 2 unicast case, i.e. keep track (via virtual queues) of overheard packets and suitably combine packets (in the spirit of ) SRC/TX 1. Create virtual queues indexed by sets where or 3. Depending on FB, we may move a packet into another queue (left-to-right manner) and use counters to keep track of how many packets a user wants from a queue. 2. Transmit linear combinations of all packets stored in queues. {1,2} {3,4} {1,2,4} {3,4,1} {1,2,3} {3,4,2} {1,2,3,4} The trick to achieve optimality is to combine more than one queues. At present, it is not clear how to extend optimal scheme to higher N. We need an exponential number of queues.

7. NETCOD 2013 June 7-9, Calgary A (suboptimal) extension to arbitrary N Scheme EXTN: Instead of combining queues, process each queue individually. Outer bound: TS inner bound: but at the cost of exponential number of queues Benefit over TS Q: can we retain the benefit over TS with a polynomial number of queues? (A: yes, with just 3 queues for any N) ΕΧΤΝ: Benefit over TS decreases vs N. The exact opposite of the multiple unicast case!

8. NETCOD 2013 June 7-9, Calgary Simplest scheme with 1-O( 2 ) performance SRC/TX If transmitted packet is erased (*) by at least one user in and received by all users in move packet from to Rule for packet movement:

9. NETCOD 2013 June 7-9, Calgary Is timesharing optimal as N ? We need to show that for any coding scheme it holds: Let be the feedback ACK/NACK sequence for user at transmission. : feedback ACK/NACK sequence for user : feedback ACK/NACK sequence for all users Define pseudo-distance: CR + memoryless + erasure: Fano: Distance:

10. NETCOD 2013 June 7-9, Calgary Conclusions For small number of users (say, <10) per group, feedback offers significant benefit, especially for high. We can construct schemes with exponential or polynomial number of queues which outperform timesharing for any finite N. For large number of users (say, >100) per group, all proposed schemes perform very close to timesharing feedback is not beneficial. There exists partial evidence that timesharing is asymptotically optimal as.

11. NETCOD 2013 June 7-9, Calgary Thank you!